Proving Quadrilateral Properties
Teacher Notes:
In this activity, students will use triangle congruence postulates and CPCTC to prove many of the
properties of special quadrilaterals.
Prerequisite: Students have already discovered properties of special quadrilaterals through a
Geometer’s Sketchpad investigation.
Students work in groups of 3-4 to complete three proofs of quadrilateral properties, then prepare a
poster of an assigned proof of a property. Each group’s assigned proofs are designed so that
students may use previously proven properties in subsequent proofs.
Then, groups will prepare a poster to present one of their property proofs as assigned by the teacher.
Each poster should include a diagram, given information, and clear steps with reasons.
On the following pages are:
 Sets of 3 proofs for each group. There are 4 different sets, so there will be some repeats
between groups. All sets have the same first proof. Given and Prove statements are
provided in both written form and as symbolic geometric statements. Diagrams are also
included. It is assumed that this is students’ first encounter turning a written sentence (e.g.
“Prove the diagonals of a rectangle are congruent”) into symbolic geometric information,
and parsing out the ‘Given’ vs. ‘Prove’ from a sentence, as well as drawing an appropriate
diagram. This can be revised to offer more challenge to students if desired.

Teacher hints for each groups proofs. This is really just a very brief summary of how
students may prove each property using Congruent Triangles.
Group A
Proving the Properties of Special Quadrilaterals
You have discovered some of the properties of special quadrilaterals in class. Today, you will use
your knowledge of Triangle Congruence Postulates and proofs to PROVE that these properties
always hold true. Do all proofs on a separate sheet.
1.
Prove that the opposite sides of a parallelogram are congruent.
Given: Parallelogram PARL,
So PA PLR and PL PAR
Prove: Opposite sides are congruent,
Or PL  AR and PA  LR
2.
A
P
R
L
Prove that the diagonals of a parallelogram bisect each other.
Given: Parallelogram PARL, or PA PLR and PL PAR
Opposite sides of PARL are congruent, so PL  AR and PA  LR
(results from #1)
A
P
Prove: Diagonals bisect each other,
Or AL bisects PR and also PR bisects AL .
M
R
L
3.
Prove that the diagonals of a rhombus bisect opposite angles.
Given: Rhombus RHOM, so RH  HO  OM  MR
Diagonals bisect each other, so RC  CO and MC  CH
(results from #2)
H
R
Prove: Diagonals bisect opposite angles,
Or MH bisects RMO and RHO
And also RO bisects MRH and MOH
C
M
4.
O
Design and make a poster presenting your proof for #
. Include the Given and
Prove, a diagram, and statements and reasons of your proof. Be prepared to explain how the
proof guarantees the quadrilateral property in question.
Group B
Proving the Properties of Special Quadrilaterals
You have discovered some of the properties of special quadrilaterals in class. Today, you will use
your knowledge of Triangle Congruence Postulates and proofs to PROVE that these properties
always hold true. Do all proofs on a separate sheet.
1.
Prove that the opposite sides of a parallelogram are congruent.
Given: Parallelogram PARL,
So PA PLR and PL PAR
Prove: Opposite sides are congruent,
Or PL  AR and PA  LR
2.
A
P
R
L
Prove that the diagonals of a parallelogram bisect each other.
Given: Parallelogram PARL, or PA PLR and PL PAR
Opposite sides of PARL are congruent, so PL  AR and PA  LR
(results from #1)
P
Prove: Diagonals bisect each other,
Or AL bisects PR and also PR bisects AL .
A
M
R
L
3.
Prove that the diagonals of a rhombus are perpendicular.
Given: Rhombus RHOM, so RH  HO  OM  MR
Diagonals bisect each other, so RC  CO and MC  CH
(results from #2)
Prove: Diagonals are perpendicular,
Or RO  MH
C
M
4.
H
R
O
Design and make a poster presenting your proof for #
. Include the Given and
Prove, a diagram, and statements and reasons of your proof. Be prepared to explain how the
proof guarantees the quadrilateral property in question.
Group C
Proving the Properties of Special Quadrilaterals
You have discovered some of the properties of special quadrilaterals in class. Today, you will use
your knowledge of Triangle Congruence Postulates and proofs to PROVE that these properties
always hold true. Do all proofs on a separate sheet.
1.
Prove that the opposite sides of a parallelogram are congruent.
Given: Parallelogram PARL,
So PA PLR and PL PAR
Prove: Opposite sides are congruent,
Or PL  AR and PA  LR
2.
A
P
R
L
Prove that the opposite angles of a parallelogram are congruent.
Given: Parallelogram PARL, or PA PLR and PL PAR
Opposite sides of PARL are congruent, so PL  AR and PA  LR
(results from #1)
A
P
Prove: Opposite angles are congruent,
or LPA  ARL and PLR  RAP
L
3.
R
Prove that the diagonals of a rectangle are congruent.
Given: Rectangle RECT, so TRE,REC,ECT,CTR  90
Opposite sides of RECT are congruent, so RE  TC and RT  EC
(results from #1)
E
R
Prove: Diagonals are congruent,
Or RC  ET
T
4.
C
Design and make a poster presenting your proof for #
. Include the Given and
Prove, a diagram, and statements and reasons of your proof. Be prepared to explain how the
proof guarantees the quadrilateral property in question.
Group D
Proving the Properties of Special Quadrilaterals
You have discovered some of the properties of special quadrilaterals in class. Today, you will use
your knowledge of Triangle Congruence Postulates and proofs to PROVE that these properties
always hold true. Do all proofs on a separate sheet.
1.
Prove that the opposite sides of a parallelogram are congruent.
Given: Parallelogram PARL,
So PA PLR and PL PAR
Prove: Opposite sides are congruent,
Or PL  AR and PA  LR
2.
A
P
R
L
Prove that one diagonal of a kite bisects opposite angles.
Given: Kite KITE,
or KI  KE and TI  TE
I
T
K
Prove: Diagonal bisects opposite angles,
or KT bisects IKE and ITE
3.
E
Prove that the diagonals of a kite are perpendicular.
Given: Kite KITE, or KI  KE and TI  TE
Diagonal bisects opposite angles, or KT bisects IKE and ITE
(Result from #2)
I
Prove: Diagonals are perpendicular,
Or IE  KT
K
M
T
E
4.
Design and make a poster presenting your proof for #
. Include the Given and
Prove, a diagram, and statements and reasons of your proof. Be prepared to explain how the
proof guarantees the quadrilateral property in question.
SUMMARIES FOR TEACHER:
Proving Opposite Sides of a P-gram are  :
Alt. Int. angles are congruent, PAL VRLA by ASA, then CPCTC.
Proving Opposite Angles of a P-gram are  :
Alt. Int. angles are congruent, PAL VRLA by ASA, then CPCTC for one set of angles.
For 2nd set of angles, repeat with PLR VRAP , or alternately, use 2 sets of adjacent angles
from PAL VRLA and angle sum.
Proving Diagonals of a P-gram bisect each other:
Alt. Int. angles are congruent, PML VRMA by ASA, then CPCTC and def’n of midpoint.
Alternately can use PMA VRML with same reasons.
*Proving Diagonals of a rhombus bisect opposite angles (TRICKY!):
Prove all 4 smaller triangles are congruent by SSS, then use CPCTC.
Alternately prove RMH VOMH by SSS, then CPCTC, then repeat with RMO VRHO .
*Proving Diagonals of a rhombus are perpendicular (TRICKY!):
Prove any 2 adjacent small triangles are congruent (ex. RMC VRHC ) by SSS, then
CPCTC to prove adjacent congruent angles, then if two angles are congruent and
supplementary, they must be right, therefore diagonals are perpendicular.
Proving Diagonals of a Rectangle are Congruent:
Prove RTC VECT (or any other pair of overlapping triangles) by SAS, then use CPCTC
to show hypotenuses are congruent.
Proving one diagonal of a kite bisects opposite angles:
Prove KIT VKET by SSS, then use CPCTC.
Proving diagonals of a kite are perpendicular:
Prove KMI VKME (or TMI VTME ) by SAS, then CPCTC to get congruent
supplementary angles.